One of the main differences between 1D problems discussed in Chapter 7 and their higher-dimensional counterparts is the presence of rotational degrees of freedom. For reasons elaborated in Chapter 5, the description of such rotational motion necessitates the introduction of the concept of angular momentum, which will be the main focus of this chapter.

General considerations

It was mentioned in §5.7.2 that the rotation algebra admits both difierential and matrix representations. In order to discuss some general properties of both kinds of representations, it is customary to denote the angular momentum generators by J. The letter L would then apply only for the difierential generators or orbital angular momenta, and the letter S will be used for spin if necessary.

To reiterate, the angular momentum algebra is the set of the following commutator relations:

In a compact notation, we can write all three relations as

where the indices take the ‘values’ x, y, z, and εαβγ is the completely antisymmetric Levi-Civita symbol defined earlier in 5.7.

In order to discuss the angular momentum states, we have to first set up a basis on the vector space. Earlier in Chapter 2, we said that the eigenstates of a Hermitian operator can be used as a basis. We can choose, for this purpose, the eigenvectors of one of the three components of the angular momentum. These states will not be eigenstates of the other two components, since the operators for difierent components of J do not commute, as seen in Eq. (8.2). Conventionally, one chooses the eigenstates of Jz. This is, of course, no special assumption: we can always consider the basis as the eigenvectors of angular momentum component in one direction and call this direction to be the z direction.

But there is a problem. There is a lot of degeneracy among the eigenstates of Jz. Whenever there is degeneracy, the eigenstates cannot be uniquely defined, because any combination of states in a degenerate subspace of eigenvectors also qualifies as an eigenvector. For a basis, we need well-defined vectors, with no ambiguity in the defintion.

In the silicon complementary metal-oxide–semiconductor (CMOS) process discussed in Chapter 2, the “back-end” (wiring) portion of the process flow was described (see Figure 11.1(a)) with tungsten (W) vias, two layers of Cu wiring and two layers of deposited dielectric. In the discussion in Chapter 2, perhaps an inkling of the actual complexity of this part of the process was given through the brief discussion of TiN or TaN barrier/adhesion layers, chemical vapor deposition (CVD) of tungsten, chemical–mechanical polishing (CMP) to planarize the W layer, deposition of a copper (Cu) seed layer, followed by a thicker electroplated Cu layer, and a “dual damascene” lithography and etching process to pattern the Cu layers. An actual image of a similar back-end silicon CMOS structure is also shown in Figure 11.1(b).

We have discussed 1D problems in Chapter 7. In this chapter, we discuss mainly 3D problems, although some of the generalities might apply to spaces whose dimensions are difierent.

Generalities

The stationary state form of the Schrödinger equation for a single particle system was given in Eq. (4.24, p. 86). For non-relativistic systems with velocity-independent potential energies, the general form of the Hamiltonian would be

for a particle of mass M. Accordingly, the Schrӧdinger equation for stationary states has the form

The Laplacian operator, ▿2, involves second-order derivatives with respect to the coordinates. This is the difierential equation we want to solve in this chapter for various choices of the potential.

Particle in a 3D box

The problem of a particle in an infinite rectangular-shaped 3D potential well is just an extension of the same problem in 1D. So, the solution can be borrowed from what we have already done in §7.2, but there will be some new features of the solutions which need some attention.

The potential is infinite except within the box defined by

Inside the box, the potential vanishes. The wavefunctions for the stationary states can be written in the form

and the difierential equation corresponding to each factor will be the same as the corresponding 1D problem.

The interface between a semiconductor and an insulator often determines the viability of the material combination in device structures. Silicon is unique in nature, at least among the semiconductors, for having a robust, reliable oxide that can be grown on its surface. The interface between Si and SiO2 is perhaps the most carefully studied of all material interfaces, and is probably the principal reason why silicon has been the dominant semiconductor material. The fact is that silicon naturally oxidizes in the sense that it can be simply placed in a furnace at high temperature with oxygen or water vapor and one obtains a nice, stable dielectric material that is essentially electrically perfect. This distinguishes silicon from all the other simple column IV semiconductor materials. Germanium can be oxidized, but its oxide is soluble in water, which makes it very hard to do any sort of chemical processing.

We have laid down the basic techniques for solving second-order homogeneous differential equations. In this appendix, we apply these techniques to some differential equations of special interest, which appeared at various parts of the text. We perform series solutions to identify the functions that emerge as solutions of these equations. Note that for some of these differential equations, the point x = 0 is an ordinary point so that two independent series solutions are possible. For some others, the point x = 0 is a regular singular point. We do not point out which differential equation belongs to which category, because it is easy to make that decision from the look of any equation.

In this chapter, we describe the simplest possible system, viz., a free particle. To keep the notation simple, we will often pretend in this chapter that the particle moves only in one spatial dimension, so that there is only one coordinate x and its corresponding momentum p. However, the notation can be trivially modified to make the discussion applicable to motion of particles in 3D space.

Solution of free Schrödinger equation

We have introduced the Schrödinger equation in Chapter 3, which says that the time evolution of the state vector is governed by the Hamiltonian of the system:

If the Hamiltonian does not have any explicit time dependence, the solution of this equation is simply

as given in Eq. (4.15, p. 85). This solution is valid irrespective of whether the particle is free. For a free particle, the extra simplification comes from the fact that the Hamiltonian involves only the momentum operator and not the position operator. Therefore, H commutes with all components of the momentum, and so the momentum eigenstates are also energy eigenstates. This means that we can use the eigenstates of the momentum operator as the energy eigenstates and write

where Ep is the energy eigenvalue that depends on the momentum eigenvalue p. This feature can also be understood as a consequence of translational invariance.

Lithography is arguably the most important process step in modern integrated circuit (IC) manufacturing. The ability to print patterns with features as small as 10–20 nm and to place those patterns on a substrate with a precision of a few nanometers is what makes today’s chips possible. Virtually all ICs are manufactured today with deep-ultraviolet (DUV) optical lithography operating with 193 nm photons, the basic process introduced in Figure 1.7.

In this chapter, we shall treat quantum mechanical problems where the system Hamiltonian either has explicit time dependence or has some parameter that is time-dependent. In either case, the Schrödinger equation is time-dependent and we are going to discuss techniques, either perturbative or exact, to solve such equations. In §12.1, we shall discuss the general formalism for treating such problems. Next, in §§ 12.2 and 12.3, we provide an analysis of the timedependent Hamiltonian for a two-level system. This will be followed by §12.4, where perturbative solutions to time-dependent Schrödinger equation shall be outlined. Finally, we analyze some aspects of Hamiltonians with periodic time dependence in §12.5.

In this chapter, we introduce some of the more popular ML algorithms. Our objective is to provide the basic concepts and main ideas, how to utilize these algorithms using Matlab, and offer some examples. In particular, we discuss essential concepts in feature engineering and how to apply them in Matlab. Support vector machines (SVM), K-nearest neighbor (KNN), linear regression, Naïve Bayes algorithm, and decision trees are introduced and the fundamental underlying mathematics is explained while using Matlab’s corresponding Apps to implement each of these algorithms. A special section on reinforcement learning is included, detailing the key concepts and basic mechanism of this third ML category. In particular, we showcase how to implement reinforcement learning in Matlab as well as make use of some of the Python libraries available online and show how to use reinforcement learning for controller design.